Frequency dependent thermal expansion in binary viscoelasticcomposites Page: 3 of 43
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are nonuniform due to nonuniform local heating and will result in complicated interactions
between thermal expansion and viscoelastic creep losses to heat.
For a viscoelastic system, there are at last two distinct and important sets of constants:
one for zero frequency (very long times) and one for infinite freugency (very short times)
(Vinogradov and Milton, 2005; Hsieh and Tuan, 2006; 2007). Therefore, much of the analysis
of viscoelastic systems can be concentrated on certain fixed frequencies (for example, f = 0
and f = oo). So we will not dwell on the details of the frequency dependence of j3* itself,
or that of the usual viscoelastic constants here, but rather concentrate on establishing the
complex nature of * thereby making the main point of the paper. Textbooks are available
(Christensen, 1982; Lakes, 1999) on the frequency dependent behavior of the viscoelastic
constants, and the time-dependent thermal expansion behavior then follows directly from
the formulas discussed here.
Methods similar to the main one used to relate overall thermal expansion coefficients
of composites to overall bulk modulus have also been used in other physical contexts by
various authors (Cribb, 1968; Schulgasser, 1989; Berryman and Milton, 1991; 1992). In
particular, the method has sometimes been called the "method of uniform fields" (Dvorak,
1990; Benveniste and Dvorak, 1990; 1992; Dvorak and Benveniste, 1992; 1997), although
it is nevertheless basically the same idea used by Cribb (1968) in the thermoelastic media
context. A similar approach applied in a much wider context has also been introduced by
Grabovsky et al. (2000).
We consider binary viscoelastic composites, and elaborate two cases: (1) composites hav-
ing overall statistically isotropic behavior, and (2) composites having hexagonal symmetry
especially those having transversely isotropic symmetry due to layering of two constituents.
2. Isotropic binary viscoelastic composites
As illustrated in Figure 1, we first consider a binary composite containing two types of
constituents labelled respectively A and B. The shape of the constituent will play no role
in the present section of the paper, but it will come into play in the next section because of
assumed layering of the components.
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Berryman, James G. Frequency dependent thermal expansion in binary viscoelasticcomposites, article, December 1, 2007; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc901856/m1/3/: accessed April 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.